From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7711 Path: news.gmane.org!not-for-mail From: Toby Bartels Newsgroups: gmane.science.mathematics.categories Subject: Re: "Terminolgy" re-visited Date: Tue, 7 May 2013 09:15:09 -0700 Message-ID: References: <91D4EC91-1FDC-45E6-9A36-7132291436C1@wanadoo.fr> Reply-To: Toby Bartels NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: 8bit X-Trace: ger.gmane.org 1367973326 32207 80.91.229.3 (8 May 2013 00:35:26 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 8 May 2013 00:35:26 +0000 (UTC) To: Categories Original-X-From: majordomo@mlist.mta.ca Wed May 08 02:35:27 2013 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.186]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1UZsLt-00016l-Vs for gsmc-categories@m.gmane.org; Wed, 08 May 2013 02:35:26 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:47044) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1UZsJx-0007ct-QH; Tue, 07 May 2013 21:33:25 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1UZsJy-0003mt-Q9 for categories-list@mlist.mta.ca; Tue, 07 May 2013 21:33:26 -0300 Content-Disposition: inline In-Reply-To: <91D4EC91-1FDC-45E6-9A36-7132291436C1@wanadoo.fr> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7711 Archived-At: Jean B?nabou wrote in part: >If F: A --> C and G: B --> C are functors I denote by F/G the comma category they define, and by F//G their 2-pull-back sometimes called their pseudo pull-back. >Answer to Toby Bartels: >(i) You propose to compose the spans by pullbacks. Here is an example of two spans A <-- X --> B and B <-- Y --> C such that the functors A <--X and Y --> C are isos, the functors X --> B and B <-- Y have unique quasi inverses, and if Z is the pullback none of the functors A <-- Z and Z --> B is a weak equivalence: >Take A=X=Y=C=1, take for B the coarse category with two objects a and b , for X --> B and B <-- Y the functors "a": 1 --> B and B <-- 1: "b" . The pullback Z is 0. By "pullback", I meant what you above call "2-pull-back" or "pseudo pull-back". Then Z is again (up to isomorphism) 1. >How stronger a counter example do you need? And using zig-zags will make the situation even worse. Using zigzags, one would compose zigzags directly and use no pullbacks. >(ii) you define equivalences by spans, not "up to anything". With this definition an equivalence between 1 and 1 is any non empty coarse category. Every non empty set determines up to isomorphism such a category. thus there are at least as many equivalences from 1 to 1 in your sense as there are non empty sets. A bit much don't you think? Indeed, so one must also define natural isomorphism of equivalences. If you have any difficulty, the answer is in Makkai's anafunctor paper: http://www.math.mcgill.ca/makkai/anafun/ >(but hurry, because after this mail there might very well be a quick revision of the article). Nothing is hidden in the nLab. If it changes, click "History" at the bottom of the page. --Toby [For admin and other information see: http://www.mta.ca/~cat-dist/ ]